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Properties

Label 2352.3.f.e
Level $2352$
Weight $3$
Character orbit 2352.f
Analytic conductor $64.087$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2352,3,Mod(97,2352)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2352.97"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2352, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2352.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,-12,0,-24,0,0,0,-24,0,0,0,0,0,0,0,48] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(64.0873581775\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4}\cdot 3 \)
Twist minimal: no (minimal twist has level 42)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + (2 \beta_{2} - \beta_1) q^{5} - 3 q^{9} - 6 q^{11} + ( - \beta_{2} - 4 \beta_1) q^{13} + (\beta_{3} - 6) q^{15} + (8 \beta_{2} - \beta_1) q^{17} + (7 \beta_{2} - \beta_1) q^{19} + (3 \beta_{3} + 12) q^{23}+ \cdots + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{9} - 24 q^{11} - 24 q^{15} + 48 q^{23} - 44 q^{25} - 44 q^{37} + 12 q^{39} - 28 q^{43} - 96 q^{51} - 240 q^{53} - 84 q^{57} - 360 q^{65} + 220 q^{67} - 312 q^{71} - 20 q^{79} + 36 q^{81} - 288 q^{85}+ \cdots + 72 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 2x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} + 4\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -3\nu^{3} \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{3} + 3\beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{3} ) / 3 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1471\) \(1765\) \(2257\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
−0.707107 + 1.22474i
0.707107 − 1.22474i
0.707107 + 1.22474i
−0.707107 − 1.22474i
0 − 1.73205i 0 − 8.36308i 0 0 0 −3.00000 0
97.2 0 − 1.73205i 0 1.43488i 0 0 0 −3.00000 0
97.3 0 1.73205i 0 − 1.43488i 0 0 0 −3.00000 0
97.4 0 1.73205i 0 8.36308i 0 0 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2352.3.f.e 4
4.b odd 2 1 294.3.c.a 4
7.b odd 2 1 inner 2352.3.f.e 4
7.c even 3 1 336.3.bh.e 4
7.d odd 6 1 336.3.bh.e 4
12.b even 2 1 882.3.c.b 4
21.g even 6 1 1008.3.cg.h 4
21.h odd 6 1 1008.3.cg.h 4
28.d even 2 1 294.3.c.a 4
28.f even 6 1 42.3.g.a ✓ 4
28.f even 6 1 294.3.g.a 4
28.g odd 6 1 42.3.g.a ✓ 4
28.g odd 6 1 294.3.g.a 4
84.h odd 2 1 882.3.c.b 4
84.j odd 6 1 126.3.n.a 4
84.j odd 6 1 882.3.n.e 4
84.n even 6 1 126.3.n.a 4
84.n even 6 1 882.3.n.e 4
140.p odd 6 1 1050.3.p.a 4
140.s even 6 1 1050.3.p.a 4
140.w even 12 2 1050.3.q.a 8
140.x odd 12 2 1050.3.q.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.3.g.a ✓ 4 28.f even 6 1
42.3.g.a ✓ 4 28.g odd 6 1
126.3.n.a 4 84.j odd 6 1
126.3.n.a 4 84.n even 6 1
294.3.c.a 4 4.b odd 2 1
294.3.c.a 4 28.d even 2 1
294.3.g.a 4 28.f even 6 1
294.3.g.a 4 28.g odd 6 1
336.3.bh.e 4 7.c even 3 1
336.3.bh.e 4 7.d odd 6 1
882.3.c.b 4 12.b even 2 1
882.3.c.b 4 84.h odd 2 1
882.3.n.e 4 84.j odd 6 1
882.3.n.e 4 84.n even 6 1
1008.3.cg.h 4 21.g even 6 1
1008.3.cg.h 4 21.h odd 6 1
1050.3.p.a 4 140.p odd 6 1
1050.3.p.a 4 140.s even 6 1
1050.3.q.a 8 140.w even 12 2
1050.3.q.a 8 140.x odd 12 2
2352.3.f.e 4 1.a even 1 1 trivial
2352.3.f.e 4 7.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(2352, [\chi])\):

\( T_{5}^{4} + 72T_{5}^{2} + 144 \) Copy content Toggle raw display
\( T_{11} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} + 72T^{2} + 144 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T + 6)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 774 T^{2} + 145161 \) Copy content Toggle raw display
$17$ \( T^{4} + 432 T^{2} + 28224 \) Copy content Toggle raw display
$19$ \( T^{4} + 342 T^{2} + 15129 \) Copy content Toggle raw display
$23$ \( (T^{2} - 24 T - 504)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 1152)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 2166 T^{2} + 423801 \) Copy content Toggle raw display
$37$ \( (T^{2} + 22 T - 167)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 4248 T^{2} + 3732624 \) Copy content Toggle raw display
$43$ \( (T^{2} + 14 T - 23)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 2952 T^{2} + 2039184 \) Copy content Toggle raw display
$53$ \( (T^{2} + 120 T + 2952)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 2448 T^{2} + 1272384 \) Copy content Toggle raw display
$61$ \( T^{4} + 1632 T^{2} + 2304 \) Copy content Toggle raw display
$67$ \( (T^{2} - 110 T - 503)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 156 T + 2556)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 19926 T^{2} + 85322169 \) Copy content Toggle raw display
$79$ \( (T^{2} + 10 T - 8687)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 17928 T^{2} + 69956496 \) Copy content Toggle raw display
$89$ \( (T^{2} + 432)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 1056 T^{2} + 112896 \) Copy content Toggle raw display
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